How do we check exactly that a distribution is involutive?
I have the following definition in my book:
A $k-$dimensional distribution $\Delta$ on a manifold $M$ is a smooth choice of a k-dimensional ...

in completing my thesis I have reached a momentary impass.
I am trying to solve an exercise given in the book "Foliations II" by Candel and Conlon. In particular, Exercise 10.4.1, and I can't seem to ...

Consider the singular Riemann surface given by the following expression:
$$z^d w^d-z^d-w^d+t=0\ ,$$
where $t$ is a parameter in $(-1,1)$ and $d$ is a positive integer greater than 2.
For $t\neq0$ the ...

I've just started to study differential geometry and I've some problems with the first definitions.
We have defined a topological manifold with boundary of dimension n as a topological space $M$ such ...

Suppose I have n orthogonal unit vectors (in Euclidean space). These unit vectors may be used to describe an n-dimensional unit hypercube. A subset of these unit vectors may be combined to make lower ...

Let $M$ be a smooth, closed, connected, oriented 3-manifold and let $f: M \rightarrow \mathbb{R}$ be a self-indexing Morse function. Since $\frac{3}{2}$ is a regular value of $f$ it follows from Morse ...

Let $M$ be a closed connected hypersurface of $n$-dimensional in $\mathbb{R}^{n+1}=\{(x^1,\cdots,x^{n+1})\}$ and let $\nu$ be a smooth unit normal vector field of $M$ at $\mathbb{R}^{n+1}$, $H$ be the ...

Let M be a differentiable manifold of dimension m and also let $\{\xi_1,\dots,\xi_m\}\subset \text{T}_pM$ be an linearly independent set of the tangent bundle of M at a certain point $p\in M$.
I have ...

In Pollack's differential topology, the proper map is defined by the preimage of every compact set is compact. Here it doesn't require the map to be continuous. However, in his following claim, to a ...

I'm currently studying some basic theory about manifolds from the book 'An introduction to manifolds' by Loring W. Tu.
The problem I have with this book is that there are very little exercises, and ...

Let $M$ be a smooth manifold in $\mathbb{R}^n$. If Lebesgue measure of $M$ is zero i.e $l(M)=0$, does it mean that volume of manifold is also zero i.e $Vol(M)=0$? Are they the same thing (volume and ...

If $M$ is a compact Riemann manifold with boundary, does it have bounded geometry, which means that the injectivity radius of the manifold is positive and every covariant derivative of the Riemannian ...

Suppose that $M$ is a smooth manifold. Is it true that the fundamental group $\pi_1(M)$ always acts on $M$? If so, how this action is defined?
EDIT: Of course I want my action to be nontrivial, say ...

I have to answer the question whether $SO(n)$ is orientable or not...Actually I have no idea - could someone help me?
I already know that $SO(n)$ is a $n(n-1)/2$-dimensional manifold, but how can I ...

I am reading a paper at the moment and I have come across two statements which I want to understand. Here is the setup:
Suppose that $G$ is a Lie group which acts on a manifold $E$ differentiably and ...

This is my first exposure to Riemann surfaces. I have studied complex analysis in an introductory course, and spent the last few weeks learning a little bit of deeper theory with Conway's Functions of ...

I am trying to prove that a topological space $(X,\mathscr{T})$ is a $0$-manifold if and only if it is a countable discrete space. In the process I have to show that there exist a homeomorphism from a ...

What is the dimension of the space of planes in $\Bbb R^3$ and how do we reach the answer?
Clarification: What I am searching for is what is the least number of parameters that I need. For example, ...

I think I managed to show this statements but I am not sure about it. Since this is common problem in differentiable manifolds I was wondering if anybody has (or may write) a solution.
Let $X$ be a ...

If $M$ is a closed aspherical 3-manifold with first fundamental group $G$, then cohomology groups of $G$ and $M$ are isomorphic because $M$ is a Eilenberg-MacLane space $K(G,1)$. In particular there ...

Consider our manifold to be $\mathbb{R}^n$ with the Euclidean metric.
In several texts that I've been reading, $\{\partial/\partial x_i\}$ evaluated at $p\in U \subset \mathbb{R}^n$ is given as the ...